If \( p^{2} + q^{2} - r^{2} = 0 \), then the value of \( p^{6} + q^{6} - r^{6} \) ÷ \( p^{2}q^{2}r^{2} \), is:
The problem asks us to find the value of an algebraic expression given a condition. We are given the condition \( p^{2} + q^{2} - r^{2} = 0 \) and need to find the value of \( \frac{p^{6} + q^{6} - r^{6}}{p^{2}q^{2}r^{2}} \). We can manipulate the given condition to simplify the numerator of the expression.
Start with the given condition:
\[ p^{2} + q^{2} - r^{2} = 0 \]Rearrange the condition to isolate \(r^2\):
\[ p^{2} + q^{2} = r^{2} \]Consider the identity for \(a+b=c\), which implies \(a^3+b^3-c^3 = -3abc\). In our case, let \(a = p^2\), \(b = q^2\), and \(c = r^2\). Since \(p^2 + q^2 = r^2\), we have \(p^2 + q^2 - r^2 = 0\). This is of the form \(a+b-c=0\).
We know that if \(a+b+c=0\), then \(a^3+b^3+c^3=3abc\). A similar identity exists for \(a+b-c=0\). If \(a+b=c\), then \(a^3+b^3 = c^3 - 3abc\). Therefore, \(a^3+b^3-c^3 = -3abc\).
Substitute \(a = p^2\), \(b = q^2\), and \(c = r^2\) into this identity:
\[ (p^{2})^{3} + (q^{2})^{3} - (r^{2})^{3} = -3(p^{2})(q^{2})(r^{2}) \]Simplify the powers:
\[ p^{6} + q^{6} - r^{6} = -3p^{2}q^{2}r^{2} \]Now, substitute this result into the expression we need to evaluate:
\[ \frac{p^{6} + q^{6} - r^{6}}{p^{2}q^{2}r^{2}} = \frac{-3p^{2}q^{2}r^{2}}{p^{2}q^{2}r^{2}} \]Cancel out the common term \(p^{2}q^{2}r^{2}\) from the numerator and denominator (assuming \(p, q, r \neq 0\), which is implied for the expression to be well-defined):
\[ \frac{-3p^{2}q^{2}r^{2}}{p^{2}q^{2}r^{2}} = -3 \]C) -3
A man invested \( \frac{1}{2} \) of his capital at 5% rate of interest per annum, \( \frac{1}{3} \) of his capital at 8% per annum and the remaining at 10% rate of interest per annum. His total income from the three investments is ₹820.00 in a year. The total capital invested is:
₹16,000.00
₹6,400.00
₹8,000.00
₹12,000.00
To find the total capital invested, we can set up an equation based on the simple interest earned from each part of the investment. We will calculate the interest from each portion of the capital and sum them up to equate to the total income given.
Let the total capital invested be \(P\).
First part of investment: Capital invested = \( \frac{1}{2} P \) Rate of interest = \(5\%\) per annum Interest from first part = \[\frac{1}{2} P \times \frac{5}{100} = \frac{5P}{200} = \frac{P}{40}\]
Second part of investment: Capital invested = \( \frac{1}{3} P \) Rate of interest = \(8\%\) per annum Interest from second part = \[\frac{1}{3} P \times \frac{8}{100} = \frac{8P}{300} = \frac{2P}{75}\]
Third part of investment: First, find the fraction of capital remaining: Remaining capital = \[1 - \left( \frac{1}{2} + \frac{1}{3} \right) = 1 - \left( \frac{3+2}{6} \right) = 1 - \frac{5}{6} = \frac{1}{6} P\] Rate of interest = \(10\%\) per annum Interest from third part = \[\frac{1}{6} P \times \frac{10}{100} = \frac{10P}{600} = \frac{P}{60}\]
Total income: The total income from all three investments is ₹820. So, we sum the interests calculated: \[ \frac{P}{40} + \frac{2P}{75} + \frac{P}{60} = 820 \] To add these fractions, find the Least Common Multiple (LCM) of 40, 75, and 60. Prime factorization: \(40 = 2^3 \times 5\) \(75 = 3 \times 5^2\) \(60 = 2^2 \times 3 \times 5\) LCM = \(2^3 \times 3 \times 5^2 = 8 \times 3 \times 25 = 24 \times 25 = 600\) Now, rewrite the equation with the common denominator: \[ \frac{15P}{600} + \frac{16P}{600} + \frac{10P}{600} = 820 \] \[ \frac{(15 + 16 + 10)P}{600} = 820 \] \[ \frac{41P}{600} = 820 \]
Solve for P: \[ P = \frac{820 \times 600}{41} \] Since \(820 = 41 \times 20\): \[ P = 20 \times 600 \] \[ P = 12000 \]
The total capital invested is ₹12,000.00.
Correct Option: D) ₹12,000.00
Select the number from among the given options that can replace the question mark (?) in the following matrix.
| 14 | 12 | 10 | 8 |
| 10 | 8 | 6 | 4 |
| 8 | 6 | 4 | 2 |
| ? | 14 | 12 | 10 |
The problem presents a matrix of numbers and asks to find the missing number. We need to identify the pattern or relationship between the numbers in rows, columns, or diagonals to determine the value of the question mark.
The matrix can be represented as:
\[ \begin{pmatrix} 14 & 12 & 10 & 8 \\ 10 & 8 & 6 & 4 \\ 8 & 6 & 4 & 2 \\ ? & 14 & 12 & 10 \end{pmatrix} \]If the pattern of subtracting 2 continues, then:
To find the first number (?), we need to reverse the pattern. If \(X - 2 = 14\), then \(X = 14 + 2\).
\[ ? = 14 + 2 = 16 \]The row-wise pattern is the most consistent.
C) 16 — Based on the consistent pattern of subtracting 2 from each number to get the next number in a row, the missing number in the fourth row is \(14 + 2 = 16\).
The problem asks us to identify the number pair that does not follow the same pattern as the other three. We need to examine the relationship between the two numbers in each pair.
From the analysis, options A, C, and D follow the pattern where the second number is the cube of the first number. Option B does not follow this pattern.
B) 7 - 27 — In this pair, \(7^3 = 343\), which is not \(27\). The number \(27\) is the cube of \(3\), not \(7\). Therefore, this pair is different from the others.
The question asks for the specific period during which the Moplah Rebellion occurred. To answer this, one needs to recall the historical timeline of significant peasant and anti-colonial movements in India.
Correct Option: D) The Moplah Rebellion took place between 1921 and 1922. This period marks the peak of the rebellion, which started in August 1921 and was largely suppressed by early 1922.
This problem involves deducing family relationships from a given set of statements. We will construct a family tree or list relationships to determine how K is related to M.
Correct Option: A) Cousin
The length of the longest pole, that could be placed in a room of dimensions 10 m, 8 m and 6 m, is:
To find the length of the longest pole that can be placed in a room, we need to calculate the length of the main diagonal of the room. A room is a cuboid, and the longest pole will stretch from one corner to the opposite corner, passing through the interior of the room. The formula for the main diagonal of a cuboid with dimensions length (\(l\)), width (\(w\)), and height (\(h\)) is given by \(D = \sqrt{l^2 + w^2 + h^2}\).
Thus, the length of the longest pole that can be placed in the room is \(10 \sqrt{2} \text{ m}\).
Correct Option: C) \(10 \times \sqrt{2} \text{ m}\) is the correct length of the space diagonal, representing the longest pole that can fit in the room.
The question asks to identify the film that received the Filmfare Award for Best Film in 2020. This requires knowledge of the Filmfare Awards for that specific year.
Correct Option: B) Gully Boy was awarded the Filmfare Award for Best Film at the 65th Filmfare Awards ceremony, which took place in February 2020. The film swept many major awards that year.
Sita took an auto from her home in Andheri (A) to go to her college in Fatehpuri (F). Rather than continuing straight on the direct road to the college that had no turns, the auto driver took a diversion after 10 km and turned right at Bandra (B) crossing, then at Colaba T-point (C) after 8 km turned left, again after covering 12 km turned left at Dalhousie Building (D) and soon after 8 km turned right at Elphinston point (E) and after covering 4 km reached Fatehpuri (F). Had the auto driver taken the direct route, how much less distance would Sita have actually travelled between the starting point and the destination?
To solve this problem, we need to calculate the total distance covered by the auto driver via the diverted route and compare it with the direct route distance. The problem states that the direct road had no turns, implying it was a straight line. The diverted route involves several turns, but the key information is that the turns (right/left) and distances covered between them form a path that eventually leads to the destination. Since the question asks for how much less distance would have been travelled on the direct route, it implies that the direct route is shorter. The specific turns (right/left) are crucial for visualizing the path, but for calculating the total distance of the diverted route, we simply sum up the individual segments. The problem also states that the direct route is a straight road, and the turns eventually lead to the same destination. This implies that the total displacement (straight-line distance) from A to F is equal to the sum of the horizontal and vertical components of the journey. However, the problem simplifies this by stating that the direct road had no turns, meaning it's a straight line, and the diverted path eventually reaches the same destination. The crucial part is that the segments of the diverted path effectively cancel out the turns in terms of net displacement along the direct path. The problem is essentially asking for the difference between the total distance travelled on the diverted path and the total distance that would have been travelled on the direct path, where the direct path is implied to be equal to the sum of the net "forward" movements in the diverted path.
The auto driver took the following segments:
Total distance on the diverted route (Diverted Distance) is the sum of these segments:
\[ \text{Diverted Distance} = 10 \, \text{km} + 8 \, \text{km} + 12 \, \text{km} + 8 \, \text{km} + 4 \, \text{km} \] \[ \text{Diverted Distance} = 42 \, \text{km} \]The problem states that the auto driver took a diversion after 10 km, turned right, then after 8 km turned left, then after 12 km turned left, then after 8 km turned right, and finally after 4 km reached Fatehpuri. The key phrase is "Rather than continuing straight on the direct road to the college that had no turns". This implies that the turns effectively cancel out in terms of net displacement perpendicular to the direct route. The direct route is the sum of the segments that contribute to forward movement towards the destination, effectively ignoring the turns that take the auto away from the direct line but eventually bring it back or parallel to it.
Let's analyze the segments that contribute to the direct path. The first segment is 10 km. The subsequent turns (right, left, left, right) with distances 8 km, 12 km, 8 km, 4 km imply a rectangular path or a path where the net displacement along the original direction is the sum of certain segments. If we consider the turns, the 8 km right turn and 8 km left turn (after 12 km) effectively cancel each other out in terms of lateral displacement if they are perpendicular to the main direction. The 12 km segment and 4 km segment are along the main direction. The problem is designed such that the direct route distance is the sum of the segments that align with the main direction of travel, which are 10 km, 12 km, and 4 km.
\[ \text{Direct Distance} = 10 \, \text{km} + 12 \, \text{km} + 4 \, \text{km} \] \[ \text{Direct Distance} = 26 \, \text{km} \]This interpretation is based on the common competitive exam problem pattern where such turns imply a path that eventually aligns with the direct route, and the 'extra' travel comes from the segments that are perpendicular to the direct path but are part of the total distance covered.
The difference is the Diverted Distance minus the Direct Distance.
\[ \text{Difference} = \text{Diverted Distance} - \text{Direct Distance} \] \[ \text{Difference} = 42 \, \text{km} - 26 \, \text{km} \] \[ \text{Difference} = 16 \, \text{km} \]Correct Option: D) 16 km
The question asks to identify the Viceroy of India during whose term the Jallianwala Bagh massacre occurred. This requires knowledge of key historical events and the corresponding administrative periods of British Viceroys in India.
Correct Option: B) Lord Chelmsford was the Viceroy of India from 1916 to 1921. The Jallianwala Bagh massacre, a pivotal event in the Indian independence movement, occurred on April 13, 1919, during his term.